Transcript Transport dans les Solides Cours de L’Ecole Doctorale (0381)
Transport in Solids Introduction
Peter M Levy New York University
A general review of the history of GMR can be found in:
http://wiki.nsdl.org/index.php/PALE:Clas sicArticles/GMR
Material I cover can be found in General:
Solid State Physics,
N.W. Ashcroft and N.D. Mermin (Holt, Rinehardt and Winston, 1976)
Electronic Transport in Mesoscopic Systems,
University Press, 1995).
S. Datta (Cambridge
Transport Phenomena,
Oxford, 1989).
H. Smith and H.H. Jensen ( Clarendon Press, J. Rammer and H. Smith, Rev. Mod. Phys.
58
, 323 (1986).
Ab-initio theories of electric transport in solid systems with reduced dimensions,
P. Weinberger, Phys. Reports
377
, 281-387 (2003).
Electrical conduction in magnetic media
How we got from 19th century concepts to applications in computer storage and memories.
1897 - The electron is discovered by J.J. Thomson
~ 1900 Drude model of conduction based on kinetic theory of gases {PV=RT}
~1928 Sommerfeld model of conduction in metals
l
~ 100
A n
n eff
~
N
(
F
)
N
(
F
)
n
F
;
k B T
or
eEl
V
IR
RI
Ohm' s law j = I A ;
E
V L
;
AR L
E
j j
nev v avg
eE
m
; is the time between collisions
j
ne
2
m
E
E
1
l
v
mean free path ~ 10 - 100A
Phenomena
While each atom scatters electrons, when they form a periodic array the atomic background only electrons from one state k to another with k+K.
This is called Bragg scattering ; it is responsible for dividing the continuous energy vs. momentum curve into bands.
n
n eff
~
N
(
F
) ;
N
(
F
)
n
F
;
k B T
or
eEl
Provides explanation for negligible contribution of conduction electrons to specific heat of metals.
What distinguishes a metal from an insulator
Magnetoresistance Lorentz force acting on trajectory of electron;longitudinal magnetoresistance (MR).
A.D. Kent Mat
.
13
et al
J. Phys. Cond.
, R461 (2001)
Anisotropic MR Role of spin-orbit coupling on electron scattering A.D. Kent Mat
.
13
et al
J. Phys. Cond.
, R461 (2001)
Domain walls
References Spin transport:
Transport properties of dilute alloys,
I. Mertig, Rep. Prog. Phys.
62,
123-142 (1999).
Spin Dependent Transport in Magnetic Nanostructures,
edited by S. Maekawa and T. Shinjo ( Taylor and Francis, 2002).
GMR:
Giant Magnetoresistance in Magnetic Layered and Granular Materials,
by P.M. Levy, in MA, 1994) pp. 367-462.
Solid State Physics
Vol
.
47,
eds. H. Ehrenreich and D. Turnbull (Academic Press, Cambridge,
Giant Magnetoresistance in Magnetic Multilayers,
by A. Barthélémy, A.Fert and F. Petroff,
Handbook of Ferromagnetic Materials,
Vol.
12
, ed. K.H.J. Buschow (Elsevier Science, Amsterdam, The Netherlands, 1999) Chap. 1.
Perspectives of Giant Magnetoresistance,
by E.Y. Tsymbal and D,G.
Pettifor, in
Solid State Physics
Vol
.
56,
eds. H. Ehrenreich and F. Spaepen (Academic Press, Cambridge, MA, 2001) pp. 113-237.
CPP-MR: M.A.M. Gijs and G.E.W. Bauer, Adv. in Phys.
46
, 285 (1997).
J. Bass, W.P. Pratt and P.A. Schroeder, Comments Cond. Mater. Phys.
18
, 223 (1998).
J. Bass and W.P. Pratt Jr., J.Mag. Mag. Mater.
200
, 274 (1999).
Spin transfer: A. Brataas, G.E.W. Bauer and P. Kelly, Physics Reports 427, 157 (2006).
Spintronics - control of current through spin of electron
The two current model of conduction in ferromagnetic metals
1988 Giant magnetoresistance Albert Fert & Peter Grünberg Parallel configuration Antiparallel configuration Two current model in magnetic multilayers
Data on GMR M.N. Baibich
et al.,
Phys. Rev. Lett.
61
, 2472 (1988).
GMR in Multilayers and Spin-Valves
1 1 0
Co 95 Fe 5 /Cu [110]
multi-layer D
R/R~110% at RT Field ~10,000 Oe
spin-valve D
R/R~8-17% at RT Field ~1 Oe
-4 0 -4 0 0 0 4 0 H / / [ 0 1 1 ]
[011] Py/Co/Cu/Co/Py NiFe Co nanolayer Cu Co nanolayer NiFe FeMn
4 0 GMR -metallic spacer between magnetic layers -current flows in plane of layers
NiFe + Co nanolayer S.S.P. Parkin
Current in the plane (CIP)-MR vs Current perpendicular to the plane (CPP)-MR
1995 GMR heads From IBM website; 1.swf
2.swf
Tunneling-MR Two magnetic metallic electrodes separated by an insulator; transport controlled by tunneling phenomena not by characteristics of conduction in metallic electrodes
2000 magnetic tunnel junctions used in magnetic random access memory From IBM website; http://www.research.ibm
.
com/research/gmr.html
PHYSICAL REVIEW LETTERS VOLUME J. A. Katine, F. J. Albert, and R. A. Buhrman E. B. Myers and D. C. Ralph
84
, 3149 (2000)
Current-Driven Magnetization Reversal and Spin-Wave Excitations in CoCuCo Pillars
School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853
Spin Accumulation from left layer
m
(
z
)
M L
z
Spin Accumulation-left layer-current reversed
m
(
z
)
M L M R
z
M R
j j How reversal in current directions changes alignment of layers
How can one rotate a magnetic layer with a spin polarized current?
By spin torques: Slonczewski-1996 Berger -1996 Waintal et al-2000 Brataas et al-2000 By current induced interlayer coupling: Heide- 2001
Current induced switching of magnetic layers by spin polarized currents can be divided in two parts: Creation of torque on background by the electric current, and reaction of background to torque.
Latter epitomized by Landau-Lifschitz equation; micromagnetics Former is current focus article in PRL:
Mechanisms of spin-polarized current-driven magnetization switching
by S. Zhang, P.M. Levy and A. Fert. Phys. Rev. Lett.
88
, 236601 (2002).
Extension of Valet-Fert to noncollinear multilayers
Methodology
To discuss transport two calculations are necessary: •Electronic structure, and •Transport equations; out of equilibrium collective electron phenomena.
Structures •Metallic multilayers •Magnetic tunnel junctions •Insulating barriers •Semiconducting barriers •Half-metallic electrodes •Semiconducting electrodes different length scales
Prepared by Carsten Heide
Lexicon of transport parameters Spin independent transport
F
Fermi energy
v F
Fermi velocity
1
k k F
mfp
Fermi momentum Mean time between collisions
mfp
Distance travelled between collisions
G
(
r v F
mfp
r
',
F
)
e i
(
k F
i
)
r
r
'
Spin dependent transport parameters
s
sf
Spin dependent relaxation time Time between spin flips
s
, /
M
,
m
sdl
sf
mfp
Spin diffusion length
d J
hv F J
Spin coherence length due to temporal precession;
tr J
exchange constant Transverse spin coherence length
J
d J
mfp
transverse spin diffusion length
l c
1
k F
k F
Transverse spin coherence length due to spatial precession.
Spin and charge accumulation in metallic systems
Derivation of Landauer formula (see Datta)
I
nev
For a conductor of length L (one dimensional electron gas) the electron density for each k state is 1/L. Thus the current carried by k states travelling in one direction is
I
e L
vf k
( )
M
( )
e L
1
k
k f
( )
M
( )
k
2(
spin
)
L
2
dk I
2
e h
d
f
( )
M
( )
I
I
( 1 )
I
( 2 ) 2
e M
( 1 2 ) 2
e
2
h h MV G c
2
e
2
h M
Therefore the contact resistance of a ballistic conductor is
G c
1 12.9
k
M
Landauer reasoned that when the conductor is not perfectly ballistic, i.e., has a transmission probability T<1 that
G
2
e
2
h MT
so that
G
1
h
1 2
e
2
M G c
1
T G s
1
h
2
e
2
M
h
2
e
2
M
1
T T
In other words when T < 1 in addition to the contact resistance there is a reistance due to the scattering in the conductor.
While the latter is independent of the length L of the conductor, it can be directly related to " ohmic" resistance as follows.
For a wide conductor W with many channels or modes of conduction M ~ W/( /k F ), so that the conductance is
G
e
2
W
(
m
/ 2 )(
v F T
(
L
) / )
How does T depend on L
?
If we neglect treat the quantum interference between electrons, the transmission probability through a conductor of length L which contains scatterers is :
T
(
L
)
L L
0
L
0 where L 0 is the average distance travelled between scatterings. This is derived as follows :
When one has a sequence of two scatterings with probabilities of transmission T 1 and T 2 , then the joint transmission probability is
not
simply the product of them; rather one have to take into account the multiple reflections :
T
1
T
2
R
1
R
2 ,
T
1
T
2
R
1 2
R
2 2 ,..... so one arrive at,
T
12 1
T
1
T
2
R
1
R
2 where
T i
1
R i
. This can be rewritten as 1-
T
12
T
12 1-
T
1
T
1 1-
T
2
T
2 , which for N scatterings in series can be extended to 1-
T
(
N
)
T
(
N
)
N
1-
T T
T
(
N
)
T N
(1
T
)
T
.
If we denote the linear
density
of scatters in a sample as scatterers in a conductor of length
L
is
N
; then the number of
L
. Placing this in the preceeding equation we find
T
(
L
)
L L
0
L
0 where
L
0
T
(1
T
) .
To identify
L
0 we note that the mean free path,
L m
, is the average distance an electron travels before being scattered; as the probability of scattering is ( 1-
T
) we can write (1
T
)
L m
~ 1
L m
1 (1
T
) ~
L
0 , when
T
~ 1.
By placing this result
T
(
L
)
L L
0
L
0 into the conductance
G
e
2
W
(
m
/ 2 )(
v F T
(
L
) / ) we find
G
e
2 (
m
/ 2 )(
v F L
0 / )
L W
L
0
G
1
L
W L
0 .
We thereby arrive at resistance made up of the combination of an actual (Ohmic), and a contact resistance :
G s
1
L
W
, and
G c
1
L
0
W
.
Conclusion In general we can always write
G
1
h
1 2
e
2
m T
h
2
e
2
m
h
2
e
2
m
1
T T
G c
1 actual resistance.
Ballistic transport: see S. Datta
Electronic Transport in Mesoscopic Systems (Cambridge Univ. Press, 1995).
Collisionless regime; transport conditions set by reservoirs at boundaries. Conductance measured by transmission through states on Fermi surface
T k
k
' '
t k
' ',
k
2 in units of the quantum of conduction 2
e
2 /
h
12.9
k
1
G
2
e
2
h MT
, where
M
is the number of channels.
Critique of the “mantra” of Landauer’s formula; see M.P. Das and F. Green, cond-mat/0304573 v1 25Apr 2003.
Application to magnetic multilayers
Semi-classical approaches to electron dynamics External fields are treated classically, while potential of periodic background is not.
r k
v n
(
k
) 1
e E
(
r
,
t
)
n
(
k
)
k
1
v n
(
k
)
c H
(
r
,
t
)
f
(
n
(
k
))
e
n
(
k
) 1 /
k B T
Validity 1 As long as one does not try to localize electron on length scale of unit cell, and wavelength of applied fields long compared to lattice constant.
eEa
,
c
2
gap
(
k
)
F
.
Diffusive transport Collisions assure local equilibrium of current;
a
mfp
however
L
, where
a
is lattice constant . Also,
mfp
phase coherence length of wavefunctions.
In the
diffusive
regime processes that occur on a length scale long with repsect to the mean free path have to be averaged, e.g., the distance traversed by an electron undergoing random scattering is
L
2 ~
mfp
(
mfp
cos ) 2 1/3
v F
2
mfp
.
By definning a diffusion constant
D
1/ 3
v F
2
mfp
we find
L
2
D
.
In this regime one can usually describe tran
sport by semi - classical Boltzmann equation. This is an equation of motion for an electron distribution function,
f
(
r
,
k
,
t
).
Simple derivation
df
(
k
,
r
,
t
)
dt
dk
k dt f
dr
r f dt
From semiclassical electron dynamics
f
(
k
,
r
,
t
)
t
:
r
v n
(
k
) 1
k
n
(
k
)
k
e E
(
r
,
t
) 1
v n
(
k
)
c
H
(
r
,
t
) and from :
k f
n
(
k
)
k
f
0
n
v n
(
k
) (
n
F
) at T = 0K.
Thus we find :
f
t
v
f
1
f
f
eE
v
(
F
)